Characterization of MCD-spaces by uniform distance-shell sizes

Prove or disprove that a finite ultrametric space with center of distances \(C(X)=\{c_0,\ldots,c_n\}\), where \(0=c_0<\cdots<c_n\), is an MCD-space of cardinality \(2^n\) if and only if every point has exactly \(b_0=1\) points at distance \(c_0\) and \(b_i=2^{i-1}\) points at distance \(c_i\) for \(1\leq i\leq n\).

Background

This conjecture proposes a precise local characterization of the global extremal condition defining MCD-spaces. The prescribed shell sizes double across successive nonzero center-of-distances values, yielding total cardinality $2^n$.

The statement is presented as a conjecture and is not established in the paper.